Introduction to the Bass Diffusion Model#

What is the Bass Model?#

The Bass diffusion model, developed by Frank Bass in 1969, is a mathematical model that describes how new products get adopted in a population over time. It’s widely used in marketing to forecast sales of new products, especially when historical data is limited or non-existent.

The model captures the entire lifecycle of product adoption, from introduction to saturation, making it a powerful tool for product planning and marketing strategy development.

The Motivation Behind the Bass Model#

Before the Bass model, companies struggled to predict the adoption patterns of new products. Traditional forecasting methods often failed because they couldn’t account for the social dynamics that drive product adoption. Frank Bass recognized that product adoption follows a distinct pattern:

Initial slow growth: When a product first launches, adoption starts slowly
Rapid acceleration: As more people adopt, word-of-mouth spreads and adoption accelerates
Eventual saturation: Eventually, the market becomes saturated and adoption slows down

The Bass model provides a mathematical framework to capture these patterns, enabling businesses to make more informed decisions about production planning, inventory management, and marketing resource allocation.

Mathematical Formulation#

The Bass model is based on a differential equation that describes the rate of adoption:

\[\frac{f(t)}{1-F(t)} = p + q F(t)\]

Where:

\(F(t)\) is the installed base fraction (cumulative proportion of adopters)
\(f(t)\) is the rate of change of the installed base fraction (\(f(t) = F'(t)\))
\(p\) is the coefficient of innovation or external influence
\(q\) is the coefficient of imitation or internal influence

The solution to this equation gives the adoption curve:

\[F(t) = \frac{1 - e^{-(p+q)t}}{1 + (\frac{q}{p})e^{-(p+q)t}}\]

The adoption rate at time \(t\) is given by:

\[f(t) = (p + q F(t))(1 - F(t))\]

Alternatively, this can be written as:

\[f(t) = \frac{(p+q)^2 \cdot e^{-(p+q)t}}{p \cdot (1+\frac{q}{p}e^{-(p+q)t})^2}\]

Key Components of the Bass Model Implementation#

The Bass model implementation in PyMC-Marketing consists of several key components:

Adopters - The number of new adoptions at time \(t\):

\[\text{adopters}(t) = m \cdot f(p, q, t)\]

Innovators - Adoptions driven by external influence (advertising, etc.):

\[\text{innovators}(t) = m \cdot p \cdot (1 - F(p, q, t))\]

Imitators - Adoptions driven by internal influence (word-of-mouth):

\[\text{imitators}(t) = m \cdot q \cdot F(p, q, t) \cdot (1 - F(p, q, t))\]

Peak Adoption Time - When the adoption rate reaches its maximum:

\[\text{peak} = \frac{\ln(q) - \ln(p)}{p + q}\]

The total number of adopters over time is the sum of innovators and imitators, which equals \(\text{adopters}(t)\). All of these components are directly implemented in the PyMC model, allowing us to analyze each aspect of the diffusion process separately.

Understanding the Relationship Between Components#

A key insight of the Bass model is how it decomposes adoption into two sources:

\[\text{adopters}(t) = \text{innovators}(t) + \text{imitators}(t)\]

At each time point:

Innovators (\(m \cdot p \cdot (1 - F(t))\)) represents new adoptions coming from people who are influenced by external factors like advertising
Imitators (\(m \cdot q \cdot F(t) \cdot (1 - F(t))\)) represents new adoptions coming from people who are influenced by previous adopters

As time progresses:

Initially, innovators dominate the adoption process when few people have adopted (\(F(t)\) is small)
Later, imitators become the primary source of new adoptions as the word-of-mouth effect grows
Eventually, both decrease as the market approaches saturation (\(F(t)\) approaches 1)

The cumulative adoption at any time point is:

\[\text{Cumulative Adoption}(t) = m \cdot F(t)\]

This means that as \(t \to \infty\), the cumulative adoption approaches the total market potential \(m\):

\[\lim_{t \to \infty} \text{Cumulative Adoption}(t) = m\]

Therefore, the Bass model provides a complete accounting of the market:

At each time point, new adopters are either innovators or imitators
Over the entire product lifecycle, all potential adopters (m) eventually adopt the product
The model tracks both the adoption rate (new adopters per time period) and the cumulative adoption (total adopters to date)

This structure enables marketers to understand not just how many people will adopt over time, but also the driving forces behind adoption at different stages of the product lifecycle.

Understanding the Key Parameters#

The model has three main parameters:

Market potential (m): Total number of eventual adopters (the ultimate market size)
Innovation coefficient (p): Measures external influence like advertising and media - typically \(0.01-0.03\)
Imitation coefficient (q): Measures internal influence like word-of-mouth - typically \(0.3-0.5\)

Parameter Interpretation#

A higher p value indicates stronger external influence (advertising, marketing)
A higher q value indicates stronger internal influence (word-of-mouth, social interactions)
The ratio q/p indicates the relative strength of internal vs. external influences
The peak of adoption occurs at time

\[t^* = \frac{\ln(q / p)}{p + q}\]

Innovators vs. Imitators#

The Bass model distinguishes between two types of adopters:

Innovators: People who adopt independently of others’ decisions, influenced mainly by mass media and external communications
- Mathematically represented as: \(\text{innovators}(t) = m \cdot p \cdot (1 - F(p, q, t))\)
Imitators: People who adopt because of social influence and word-of-mouth from previous adopters
- Mathematically represented as: \(\text{imitators}(t) = m \cdot q \cdot F(p, q, t) \cdot (1 - F(p, q, t))\)

Real-World Applications#

The Bass model has been successfully applied to forecast the adoption of various products and technologies:

Consumer durables: TVs, refrigerators, washing machines
Technology products: Smartphones, computers, software
Pharmaceutical products: New drugs and treatments
Entertainment products: Movies, games, streaming services
Services and subscriptions: Banking services, subscription plans

Business Value: Why the Bass Model Matters to Executives and Marketers#

From a business perspective, the Bass diffusion model provides substantial competitive advantages and ROI benefits:

1. Resource Optimization and Cash Flow Management#

Production Planning: Avoid costly overproduction or stockouts by accurately forecasting demand curves
Marketing Budget Allocation: Optimize spending across the product lifecycle, investing more during key inflection points
Supply Chain Efficiency: Coordinate with suppliers and distributors based on predicted adoption rates
Cash Flow Optimization: Better predict revenue streams, improving financial planning and investor relations

2. Strategic Decision Making#

Launch Timing: Determine the optimal time to enter a market based on diffusion patterns
Pricing Strategy: Implement dynamic pricing strategies aligned with the adoption curve
Competitive Analysis: Compare your product’s adoption parameters with competitors to identify strengths and weaknesses
Product Portfolio Management: Make informed decisions about when to phase out older products and introduce new ones

3. Risk Mitigation#

Scenario Planning: Test different market assumptions and external factors through parameter variations
Early Warning System: Identify deviations from expected adoption curves early, enabling faster intervention
Investment Justification: Provide data-driven forecasts to justify R&D and marketing investments to stakeholders

4. Performance Measurement#

Marketing Effectiveness: Measure the impact of marketing campaigns on the innovation coefficient (p)
Word-of-Mouth Strength: Quantify the power of your brand’s social influence through the imitation coefficient (q)
Total Market Potential: Validate or adjust your total addressable market estimates (m)

In today’s data-driven business environment, companies that effectively utilize models like Bass diffusion gain a significant competitive edge through more precise forecasting, better resource allocation, and strategic market timing.

Bayesian Extensions#

In this notebook, we show how to generate simulated data from the Bass model and fit a Bayesian model to it. The Bayesian formulation offers several advantages:

Uncertainty quantification through prior distributions on parameters
Hierarchical modeling for multiple products or markets
Incorporation of expert knowledge through informative priors
Full probability distributions for future adoption forecasts

What we’ll do in this notebook#

In this notebook, we’ll:

Set up parameters for a Bass model simulation
Generate simulated adoption data for multiple products
Fit the Bass model to our simulated data using PyMC
Visualize the adoption curves

Prepare Notebook#

from typing import Any

import arviz as az
import matplotlib.pyplot as plt
import numpy as np
import numpy.typing as npt
import pandas as pd
import pymc as pm
import xarray as xr
from pymc_extras.prior import Prior, Scaled

from pymc_marketing.bass.model import create_bass_model
from pymc_marketing.plot import plot_curve

az.style.use("arviz-darkgrid")
plt.rcParams["figure.figsize"] = [12, 7]
plt.rcParams["figure.dpi"] = 100

%load_ext autoreload
%autoreload 2
%config InlineBackend.figure_format = "retina"

/Users/will/mamba/envs/pymc-marketing-dev/lib/python3.11/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
  from .autonotebook import tqdm as notebook_tqdm

seed: int = sum(map(ord, "bass"))
rng: np.random.Generator = np.random.default_rng(seed=seed)

Setting Up Simulation Parameters#

First, we’ll set up the parameters for our simulation. This includes:

The time period for our simulation (in weeks)
The number of products to simulate
Start dates for the simulation period

def setup_simulation_parameters(
    n_weeks: int = 52,
    n_products: int = 9,
    start_date: str = "2023-01-01",
    cutoff_start_date: str = "2023-12-01",
) -> tuple[
    npt.NDArray[np.int_],
    pd.DatetimeIndex,
    pd.DatetimeIndex,
    list[str],
    pd.Series,
    dict[str, Any],
]:
    """Set up initial parameters for the Bass diffusion model simulation.

    Parameters
    ----------
    n_weeks : int
        Number of weeks to simulate
    n_products : int
        Number of products to include in the simulation
    start_date : str
        Starting date for the simulation period
    cutoff_start_date : str
        Latest possible start date for products

    Returns
    -------
    T : numpy.ndarray
        Time array (weeks)
    possible_dates : pandas.DatetimeIndex
        All dates in the simulation period
    possible_start_dates : pandas.DatetimeIndex
        Possible start dates for products
    products : list
        List of product names
    product_start : pandas.Series
        Start date for each product
    coords : dict
        Coordinates for PyMC model
    """
    # Set a seed for reproducibility
    seed = sum(map(ord, "bass"))
    rng = np.random.default_rng(seed)

    # Create time array and date range
    T = np.arange(n_weeks)
    possible_dates = pd.date_range(start_date, freq="W-MON", periods=n_weeks)
    cutoff_start_date = pd.to_datetime(cutoff_start_date)
    cutoff_start_date = cutoff_start_date + pd.DateOffset(weeks=1)
    possible_start_dates = possible_dates[possible_dates < cutoff_start_date]

    # Generate product names and random start dates
    products = [f"P{i}" for i in range(n_products)]
    product_start = pd.Series(
        rng.choice(possible_start_dates, size=len(products)),
        index=pd.Index(products, name="product"),
    )

    coords = {"T": T, "product": products}
    return T, possible_dates, possible_start_dates, products, product_start, coords

Creating Prior Distributions#

For our Bayesian Bass model, we need to specify prior distributions for the key parameters:

m (market potential): How many units can potentially be sold in total
p (innovation coefficient): Rate of adoption from external influences
q (imitation coefficient): Rate of adoption from internal/social influences
likelihood: The probability distribution that models the observed adoption data

For the market potential m we use a scaling trick to specify a scale-free prior and then add a global factor:

def create_bass_priors(factor: float) -> dict[str, Prior | Scaled]:
    """Define prior distributions for the Bass model parameters.

    Returns
    -------
    dict
        Dictionary of prior distributions for m, p, q, and likelihood

    Notes
    -----
    - m: Market potential (scaled Gamma distribution)
    - p: Innovation coefficient (Beta distribution)
    - q: Imitation coefficient (Beta distribution)
    - likelihood: Observation model (Negative Binomial)
    """
    return {
        # We use a scaled Gamma distribution for the market potential.
        "m": Scaled(Prior("Gamma", mu=1, sigma=0.1, dims="product"), factor=factor),
        "p": Prior("Beta", mu=0.03, dims="product").constrain(lower=0.01, upper=0.03),
        "q": Prior("Beta", dims="product").constrain(lower=0.3, upper=0.5),
        "likelihood": Prior(
            "NegativeBinomial",
            n=Prior("InverseGamma", alpha=8, beta=20),
            dims="product",
        ),
    }

Let’s generate and visualize the priors.

FACTOR = 50_000
priors = create_bass_priors(factor=FACTOR)

/Users/will/mamba/envs/pymc-marketing-dev/lib/python3.11/site-packages/preliz/distributions/beta.py:124: RuntimeWarning: invalid value encountered in scalar divide
  mu = alpha / alpha_plus_beta
/Users/will/mamba/envs/pymc-marketing-dev/lib/python3.11/site-packages/preliz/distributions/beta.py:125: RuntimeWarning: invalid value encountered in scalar divide
  sigma = (alpha * beta) ** 0.5 / alpha_plus_beta / (alpha_plus_beta + 1) ** 0.5
/Users/will/github/pymc-eco/pymc-marketing/pymc_marketing/prior.py:929: UserWarning: 
The requested mass is 0.95,
but the computed one is 0.528
  new_parameters = maxent(

fig, ax = plt.subplots(nrows=2, ncols=1, figsize=(15, 12))

priors["p"].preliz.plot_pdf(ax=ax[0])
ax[0].set(title="Innovation Coefficient (p)")
priors["q"].preliz.plot_pdf(ax=ax[1])
ax[1].set(title="Imitation Coefficient (q)")
fig.suptitle(
    "Prior Distributions for Bass Model Parameters",
    fontsize=18,
    fontweight="bold",
    y=0.95,
);

../../_images/ec8e7c76a11f86680a152e7946f8d5a3307d05f99c77d31d08d82b322e004feb.png

Observe we have chosen the priors within the usual ranges of empirical studies:

Innovation coefficient (p): Measures external influence like advertising and media - typically \(0.01-0.03\)
Imitation coefficient (q): Measures internal influence like word-of-mouth - typically \(0.3-0.5\)

Generate Synthetic Data#

With the generative Bass model, we can generate a synthetic dataset by sampling from the prior and choosing one particular sample to use as observed data. For this purpose we define two auxiliary functions.

def sample_prior_bass_data(model: pm.Model) -> xr.DataArray:
    """Generate a sample from the prior predictive distribution of the Bass model.

    Parameters
    ----------
    model : pymc.Model
        The PyMC model to sample from

    Returns
    -------
    xarray.DataArray
        Simulated adoption data
    """
    with model:
        idata = pm.sample_prior_predictive(random_seed=rng)
    return idata["prior"]["y"].sel(chain=0, draw=0)


def transform_to_actual_dates(bass_data, product_start, possible_dates) -> pd.DataFrame:
    """Transform simulation data from time index to calendar dates.

    Parameters
    ----------
    bass_data : xarray.DataArray
        Simulated bass model data
    product_start : pandas.Series
        Start date for each product
    possible_dates : pandas.DatetimeIndex
        All dates in the simulation period

    Returns
    -------
    pandas.DataFrame
        Adoption data with actual calendar dates
    """
    bass_data = bass_data.to_dataset()
    bass_data["product_start"] = product_start.to_xarray()

    df_bass_data = (
        bass_data.to_dataframe().drop(columns=["chain", "draw"]).reset_index()
    )
    df_bass_data["actual_date"] = df_bass_data["product_start"] + pd.to_timedelta(
        7 * df_bass_data["T"], unit="days"
    )

    return (
        df_bass_data.set_index(["actual_date", "product"])
        .y.unstack(fill_value=0)
        .reindex(possible_dates, fill_value=0)
    )

Now we can generate the observed data:

# Setup simulation parameters
T, possible_dates, _, products, product_start, coords = setup_simulation_parameters()

# Create and configure the Bass model
generative_model = create_bass_model(t=T, coords=coords, observed=None, priors=priors)

# Sample and select one "observed" dataset.
bass_data = sample_prior_bass_data(generative_model)
actual_data = transform_to_actual_dates(bass_data, product_start, possible_dates)

Sampling: [m_unscaled, p, q, y, y_n]

The actual_data data frame has the typical format of a real dataset.

actual_data

product	P0	P1	P2	P3	P4	P5	P6	P7	P8
2023-01-02	0	0	797	0	0	0	0	0	0
2023-01-09	0	0	9902	0	0	0	0	0	0
2023-01-16	0	0	5468	0	0	0	0	0	0
2023-01-23	0	0	4941	0	0	0	0	0	0
2023-01-30	0	1281	9015	0	0	1397	0	0	0
2023-02-06	0	3439	3036	0	0	1856	0	0	0
2023-02-13	0	3901	5675	0	0	1908	0	0	0
2023-02-20	0	3235	8003	0	0	3636	0	0	0
2023-02-27	0	1618	3953	1462	0	8804	0	0	0
2023-03-06	0	2863	2269	921	0	2615	0	0	0
2023-03-13	761	14140	1000	2080	0	7206	0	0	0
2023-03-20	1738	5093	1318	1746	0	2122	0	0	0
2023-03-27	3923	5720	1101	666	0	9001	0	0	0
2023-04-03	2984	4892	259	3487	0	8587	0	0	0
2023-04-10	3715	10664	626	4902	0	1432	0	0	0
2023-04-17	3293	1241	170	3783	0	331	0	0	0
2023-04-24	5699	2082	251	12631	0	2839	0	0	0
2023-05-01	2452	1987	106	1654	0	894	0	0	0
2023-05-08	4322	1207	61	2982	0	583	0	0	0
2023-05-15	3375	46	11	1005	0	1750	0	0	0
2023-05-22	1344	174	10	796	0	174	0	0	0
2023-05-29	1322	195	9	1072	0	319	0	0	0
2023-06-05	1593	185	11	789	0	492	0	0	0
2023-06-12	2609	195	0	397	0	134	0	0	0
2023-06-19	1078	10	4	786	0	8	0	0	0
2023-06-26	626	16	1	177	0	62	0	0	0
2023-07-03	240	29	1	213	0	22	0	0	0
2023-07-10	136	16	0	101	0	97	0	0	0
2023-07-17	106	12	0	53	0	3	0	0	0
2023-07-24	150	3	1	48	0	13	0	0	1104
2023-07-31	37	6	0	14	0	7	0	0	773
2023-08-07	17	5	1	7	0	5	0	0	2788
2023-08-14	18	1	0	7	0	1	0	0	1550
2023-08-21	5	2	1	15	0	5	0	0	2083
2023-08-28	5	1	0	3	0	0	0	0	6172
2023-09-04	14	0	0	0	0	1	0	286	1638
2023-09-11	4	0	0	3	0	1	1259	1759	9292
2023-09-18	1	0	0	0	0	0	3048	153	3218
2023-09-25	1	0	0	0	0	1	5245	2837	2360
2023-10-02	0	0	0	0	0	0	3351	1574	2411
2023-10-09	0	0	0	1	0	0	4058	9728	1869
2023-10-16	3	0	0	0	0	0	4129	4594	4091
2023-10-23	0	0	0	0	0	0	2957	3812	1734
2023-10-30	0	0	0	0	1205	0	7700	755	432
2023-11-06	0	0	0	0	349	0	2875	1755	746
2023-11-13	0	0	0	0	802	0	3873	258	82
2023-11-20	0	0	0	0	1037	0	6791	2594	790
2023-11-27	0	0	0	0	3913	0	1119	870	328
2023-12-04	0	0	0	0	2865	0	1488	2182	356
2023-12-11	0	0	0	0	5758	0	517	434	179
2023-12-18	0	0	0	0	1080	0	392	956	100
2023-12-25	0	0	0	0	3338	0	383	684	103

On the other hand, the bass_data has the same data as arrays indexed by time (relative) and product.

Let’s visualize both.

fig, ax = plt.subplots(
    nrows=2, ncols=1, figsize=(15, 12), sharex=False, sharey=True, layout="constrained"
)

# Plot raw simulated data (by time step)
bass_data.to_series().unstack().plot(ax=ax[0])
ax[0].legend(
    title="Product", title_fontsize=14, loc="center left", bbox_to_anchor=(1, 0.5)
)
ax[0].set(
    title="Simulated Weekly Adoption by Product (Time Steps)",
    xlabel="Time Step (Weeks)",
    ylabel="Number of Adoptions",
)

# Plot data with actual calendar dates
actual_data.plot(ax=ax[1])
ax[1].legend(
    title="Product", title_fontsize=14, loc="center left", bbox_to_anchor=(1, 0.5)
)
ax[1].set(
    title="Simulated Weekly Adoption by Product (Calendar Dates)",
    xlabel="Date",
    ylabel="Number of Adoptions",
)

fig.suptitle(
    "Bass Diffusion Model - Simulated Product Adoption", fontsize=18, fontweight="bold"
);

../../_images/75418c4f7d50d90a8874201e3b86c3704aa82a333a2990283a40cde77081b835.png

Fit the Model#

We are now ready to fit the model and generate the posterior predictive distributions.

# We condition the model on observed data.
with pm.observe(generative_model, {"y": bass_data.values}) as model:
    idata = pm.sample(
        tune=1_500,
        draws=2_000,
        chains=4,
        nuts_sampler="nutpie",
        compile_kwargs={"mode": "NUMBA"},
        random_seed=rng,
    )

    idata.extend(
        pm.sample_posterior_predictive(
            idata, model=model, extend_inferencedata=True, random_seed=rng
        )
    )

Sampler Progress

Total Chains: 4

Active Chains: 0

Finished Chains: 4

Sampling for now

Estimated Time to Completion: now

Draws	Step Size	Gradients/Draw
3500	0.47	15
3500	0.46	15
3500	0.46	15
3500	0.47	15

Sampling: [y]

/Users/will/mamba/envs/pymc-marketing-dev/lib/python3.11/site-packages/rich/live.py:231: UserWarning: install 
"ipywidgets" for Jupyter support
  warnings.warn('install "ipywidgets" for Jupyter support')

We do not have any divergences. Let’s look at the summary of the parameters.

az.summary(data=idata, var_names=["p", "q", "m", "y_n"])

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
p[P0]	0.028	0.005	0.019	0.037	0.000	0.000	11763.0	6085.0	1.0
p[P1]	0.027	0.004	0.019	0.035	0.000	0.000	12486.0	5589.0	1.0
p[P2]	0.039	0.006	0.029	0.050	0.000	0.000	12658.0	6225.0	1.0
p[P3]	0.026	0.005	0.017	0.034	0.000	0.000	11811.0	6400.0	1.0
p[P4]	0.027	0.005	0.018	0.036	0.000	0.000	11724.0	5613.0	1.0
p[P5]	0.027	0.005	0.019	0.036	0.000	0.000	10657.0	5906.0	1.0
p[P6]	0.031	0.005	0.022	0.041	0.000	0.000	12747.0	6105.0	1.0
p[P7]	0.027	0.005	0.019	0.037	0.000	0.000	10512.0	6201.0	1.0
p[P8]	0.025	0.004	0.017	0.033	0.000	0.000	10473.0	6146.0	1.0
q[P0]	0.409	0.018	0.374	0.443	0.000	0.000	12644.0	6048.0	1.0
q[P1]	0.410	0.017	0.379	0.443	0.000	0.000	13197.0	5887.0	1.0
q[P2]	0.441	0.019	0.404	0.476	0.000	0.000	13067.0	6615.0	1.0
q[P3]	0.411	0.018	0.376	0.446	0.000	0.000	12509.0	5858.0	1.0
q[P4]	0.371	0.017	0.336	0.401	0.000	0.000	11284.0	6026.0	1.0
q[P5]	0.377	0.017	0.346	0.408	0.000	0.000	10929.0	5561.0	1.0
q[P6]	0.433	0.019	0.398	0.468	0.000	0.000	12212.0	6178.0	1.0
q[P7]	0.318	0.015	0.290	0.347	0.000	0.000	10307.0	6402.0	1.0
q[P8]	0.352	0.016	0.322	0.382	0.000	0.000	10306.0	6106.0	1.0
m[P0]	48556.666	4129.629	40771.362	56211.078	36.046	50.583	13278.0	5763.0	1.0
m[P1]	53546.238	4234.743	45501.402	61293.719	37.407	55.199	12877.0	5654.0	1.0
m[P2]	51913.351	4321.060	44099.071	60142.530	38.732	52.322	12483.0	6086.0	1.0
m[P3]	47886.923	4296.611	39686.264	55695.049	36.339	54.168	14500.0	5468.0	1.0
m[P4]	44249.734	4084.189	36633.565	51873.099	35.007	54.048	13747.0	5440.0	1.0
m[P5]	52106.801	4188.603	44024.189	59633.874	37.839	50.806	12634.0	5463.0	1.0
m[P6]	50151.135	4340.068	41981.008	58256.426	36.599	55.887	14219.0	6010.0	1.0
m[P7]	46240.943	4131.518	38276.280	53690.682	34.373	53.147	14403.0	6077.0	1.0
m[P8]	48936.249	4105.054	41167.827	56687.960	35.481	50.344	13442.0	5903.0	1.0
y_n	2.593	0.224	2.180	3.028	0.002	0.003	10784.0	6250.0	1.0

_ = az.plot_trace(
    data=idata,
    var_names=["p", "q", "m", "y_n"],
    compact=True,
    backend_kwargs={"figsize": (12, 7), "layout": "constrained"},
)
plt.gcf().suptitle("Model Trace", fontsize=16);

../../_images/3c35bf584b4c69188dcad617ee1ece7a15a401c082076f2b1086d01decdf15c0.png

Overall, the diagnostics and trace look good.

Next, we look into the posterior distributions of the parameters.

ax, *_ = az.plot_forest(idata["posterior"]["p"], combined=True)
ax.axvline(x=priors["p"].parameters["mu"], color="gray", linestyle="--")
ax.get_figure().suptitle("Innovation Coefficient (p)", fontsize=18, fontweight="bold")

Text(0.5, 0.98, 'Innovation Coefficient (p)')

../../_images/9cdc3157fd33d9f3dbe4605af2c923fd62cf6a7060dfb72d71b465e6b15515b7.png

ax, *_ = az.plot_forest(idata["posterior"]["q"], combined=True)
ax.axvline(x=priors["q"].preliz.mean(), color="gray", linestyle="--")
ax.get_figure().suptitle("Imitation Coefficient (q)", fontsize=18, fontweight="bold")

Text(0.5, 0.98, 'Imitation Coefficient (q)')

../../_images/8937968db2b06053dbb7eb499caec52aa226a505d214f960f7744c9328959011.png

We do see some heterogeneity in the parameters, but overall they are centered around the true values (from the generative model).

Examining Posterior Predictions for Specific Products#

Let’s look at the posterior predictive distributions to see how well our model captures the simulated data.

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior_predictive"]["y"].pipe(plot_curve, {"T"}, axes=axes)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i], color="black")

fig.suptitle("Posterior Predictive vs Observed Data", fontsize=18, fontweight="bold");

../../_images/f0fdde73296a9fc4857d5ab2975ebb634a20b74521e5094af219ff02f84d6858.png

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior_predictive"]["y"].cumsum(dim="T").pipe(plot_curve, {"T"}, axes=axes)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i].cumsum(), color="black")

fig.suptitle(
    "Cumulative Posterior Predictive vs Cumulative Observed Data",
    fontsize=18,
    fontweight="bold",
);

../../_images/b0bb174ee536e75d3384648cd3f510fff804c4a3f2ef676ce94915fa50ea13e0.png

observed_cumulative = bass_data.cumsum(dim="T").isel(T=-1).to_series()

ref_val = {
    "m": [
        {"product": name, "ref_val": value}
        for name, value in observed_cumulative.items()
    ]
}

az.plot_posterior(
    idata.posterior,
    var_names=["m"],
    backend_kwargs=dict(sharex=True, layout="constrained", figsize=(15, 12)),
    ref_val=ref_val,
)

max_T = bass_data.coords["T"].max().item()
fig = plt.gcf()
fig.suptitle(
    f"Estimated Market Cap (m) vs Observed Cumulative at T={max_T}",
    fontsize=18,
    fontweight="bold",
);

../../_images/2d7827277cf509fbffdb94d0e964b7b8442d1654b045313677b2309f4040e881.png

Overall, the model does a good job of capturing the data.

Next, we look into the adopters, which represent the expected value of the likelihood.

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior"]["adopters"].pipe(plot_curve, {"T"}, axes=axes)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i], color="black")

fig.suptitle("Adopters vs Observed Data", fontsize=18, fontweight="bold");

../../_images/4c304106960d202d7fba22548012cc661b6bca2ea30ebba295b83f0f04dc41b3.png

This show the fit is indeed quite reasonable.

We can also evaluate the model goodness by looking into the cumulative data:

Note

Remember that the adopters is the mean of the distribution so we see some cumulative curves above and some below.

Look at the idata["posterior_predictive"]["y"] for the observed data.

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior"]["adopters"].cumsum(dim="T").pipe(plot_curve, {"T"}, axes=axes)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i].cumsum(), color="black")

fig.suptitle("Adopters Cumulative vs Observed Data", fontsize=18, fontweight="bold");

../../_images/fe88cd44f060c3e4ee4ea12bcf29d98d2a28efbb9bb32a7d6ef618f832c684bd.png

We can enhance this view by looking into the components of the model: innovators and imitators (in orange and green, respectively).

fig, axes = plt.subplots(
    nrows=3, ncols=3, figsize=(15, 12), sharex=True, sharey=True, layout="constrained"
)

idata["posterior"]["adopters"].cumsum(dim="T").pipe(
    plot_curve, {"T"}, colors=3 * 3 * ["C0"], axes=axes
)

idata["posterior"]["innovators"].pipe(
    plot_curve, {"T"}, colors=3 * 3 * ["C1"], axes=axes
)
idata["posterior"]["imitators"].pipe(
    plot_curve, {"T"}, colors=3 * 3 * ["C2"], axes=axes
)

for i, ax in enumerate(axes.flatten()):
    ax.plot(T, bass_data[:, i].cumsum(), color="black")

fig.suptitle("Innovators vs Imitators", fontsize=18, fontweight="bold");

../../_images/28e35d5fd283c55913baf4891ae5960549286a4af1a7f785f91b48543ec04f2e.png

Finally, we can inspect the peak of the adoption curve.

ax, *_ = az.plot_forest(idata["posterior"]["peak"], combined=True)
ax.get_figure().suptitle("Peak", fontsize=18, fontweight="bold");

../../_images/971b27e474cabfc896864e0a3a0a0b2ed93066a17e9053fd0f6e089a89adf4cc.png

This fits the observed data quite well. Let’s see for example the product P4.

fig, ax = plt.subplots()

product_id = 4

bass_data[:, product_id].plot(ax=ax, color="black")

idata["posterior"]["adopters"].sel(product=f"P{product_id}").pipe(
    plot_curve, {"T"}, axes=ax
)

peak_hdi = az.hdi(idata["posterior"]["peak"].sel(product=f"P{product_id}"))["peak"]
ax.axvspan(
    peak_hdi.sel(hdi="lower").item(),
    peak_hdi.sel(hdi="higher").item(),
    color="C1",
    alpha=0.4,
)

ax.set_title(f"Peak Product {products[product_id]}", fontsize=18, fontweight="bold");

../../_images/d685ec8287a2c8d79af5c2ee8fa4949828c01e01fc71e1402b2f29266580f096.png

%load_ext watermark
%watermark -n -u -v -iv -w -p nutpie,pymc_marketing,pytensor

Last updated: Thu Jul 24 2025

Python implementation: CPython
Python version       : 3.11.13
IPython version      : 9.3.0

nutpie        : 0.15.1
pymc_marketing: 0.15.1
pytensor      : 2.31.7

numpy         : 2.2.6
pandas        : 2.3.0
arviz         : 0.21.0
xarray        : 2025.6.1
pymc_marketing: 0.15.1
matplotlib    : 3.10.3
pymc          : 5.24.0

Watermark: 2.5.0